To fix notation, let \(A\)  be \(k \times k\)  and \(D\)  be \((n-k) \times (n-k)\) . We will follow Fischer's proof,
which involves the \(k^{\textrm {th}}\)  compound of \(G\) . This is the matrix with rows and columns
indexed by the distinct \(k\) -subsets of \(\lbrace 1, \ldots , n\rbrace \)  with the entry in row \(X\)  and
column \(Y\)  the minor \(G_{X, Y}\)  of \(G\) 
with rows labelled by \(X\)  and columns labelled by \(Y\) . We denote the \(k^{\textrm {th}}\)  compound of a matrix \(M\)  by \(M^{(k)}\) . The following results on compounds would have been well known to Fischer's contemporaries (for further discussion, see, for example, Section 0.8 of [1]}):
